Question

98 POINTS! PLEASE HELP! I been stuck on this assignment for over 2 weeks. My teacher gave me a simple explanation when I asked for help, but I still didn't understand it. I asked for more help and now they aren't responding anymore. I'm desperate. Nobody else I have asked understands it either. Here is the question:
Thomas wants to make a box with no lid out of a cardboard sheet to hold a marble collection. He plans to cut out four congruent squares from all corners of the sheet and then fold up the sides to make the box.


Thomas buys a cardboard sheet that is 8 by 12 inches. Let x be the side length of each cutout. Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function.

What is the size of the cutout he needs to make so that he can fit the most marbles in the box?

If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be? What would be the dimensions of this box?


This has to do with polynomials by factoring.


Please EXPLAIN IT so that I UNDERSTAND it. Don't just give me the answer. THANK YOU so much in advance. :)

Answer 1

See the attached picture.

• Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function.

The resulting box has a volume

$V(x)=x(8-x)(12-x)=x^3-20x^2+96x$

because the volume of a box is the product of its width $(12-x)$, length $(8-x)$, and height $(x)$.

• find the zeroes

You know right away from the factored form of $V(x)$ that the zeroes are $x=0,8,12$. (zero product property)

• sketch the graph of the function

Easy to plot by hand. You know the zeroes, and you can check the sign of $V(x)$ for any values of $x$ between these zeros to get an idea of what the graph of $V(x)$ looks like. See the second attached picture.

Here's what I mean by "check the sign" in case you don't follow. We know $V(x)=0$ when $x=0$ and $x=8$. So we pick some value of $x$ between them, say $x=1$, and find that

$V(1)=1(8-1)(12-1)=7\cdot11=77$

which is positive, so $V(x)$ will be positive for any other $x$ between 0 and 8. Similarly we would find that $V(x)$ for $x$ between 8 and 12, and so on.

• What is the size of the cutout he needs to make so that he can fit the most marbles in the box?

It's impossible to answer this without knowing the volume of each marble...

• If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be?

Thomas wants $V(x)=12$, so you solve

$x^3-20x^2+96x=12$

While this is possible to do by hand, the procedure is tedious (look up "solving the cubic equation"). With a calculator, you'd find three approximate solutions

$x\approx0.1284$

$x\approx7.6398$

$x\approx12.2318$

but you throw out the third solution because, realistically, the cutout length can't be greater than either of the sheet's dimensions.

• What would be the dimensions of this box?

The box's dimensions are ($x$ in) x ($8-x$ in) x ($12-x$ in).

If $x\approx0.1284$, then $8-x\approx7.8716$ and $12-x\approx11.8716$.

If $x\approx7.6398$, then $8-x\approx0.3602$ and $12-x\approx4.3602$.